Reorganizing freshman business mathematics II: authentic assessment in mathematics through professional memos

The first part of this two-part paper (13) described the developmentof a new freshman business mathematics (FBM) course at our college.In this paper, we discuss our assessment tool, the businessmemo, as a venue for students to apply mathematical skills,via mathematical modelling, to realistic business problems.These memos have proven a crucial step in turning our FBM coursearound from a dreaded course with little connection to students’intended careers into a course where students experience thepower of mathematics for solving problems and informing decisions.Comments from students in the course throughout its 6-year historyclearly point to the course's value and importance.     

German undergraduate mathematics enrolment numbers: background and change

Before we consider the German tertiary system, we review the education system and consider other relevant background details. We then concentrate on the tertiary system and observe that the mathematical enrolments are keeping up with the overall student enrolments. At the same time, the first year mathematics enrolments for women are greater than that for men, although more men are still studying mathematics at university. Finally, we note that the German economy seems to play a role in mathematics enrolments though not necessarily to its comparative detriment.     

Kaizen teaching and the learning habits of engineering students in a freshman mathematics course

Which are the teaching methods that actually contribute to the learning of mathematics? The answer to this certainly is the holy grail of didactic and pedagogy, and should be supported by large scale statistical evidence. Our article aims at providing an initial step into this direction by first illustrating a teaching paradigm that is suited for the generation of large scale data sets: based on industry best practice quality assurance standards we introduce the Kaizen teaching paradigm which enforces Kolb’s reflective learning cycle on the students’ side. Second, we present and analyze the data we obtained through our pilot implementation at a engineering freshman mathematics course in the Sultanate of Oman. These emphasize the effectiveness of Kaizen teaching and once again show the necessity of continuous learning. A practice that seems to be forgotten in traditional university engineering courses due to the mere size of the audience. In particular it seems that a Markovian estimator for students’ performance may have to be considered.     

Mathematics and philosophy (on the dialectical development of mathematics)

The interplay and synthesis of antithetic elements of cognitive activity of mathematicians is considered. The emphasis is on basic antithetic elements such as logic and intuition, generality and individuality, synthesis and construction. Besides, we perform a dialectical analysis of the appearance of new abstract concepts based directly on one’s sensory experience or on abstract concepts that have already been used earlier in the solution of mathematical problems. All conclusions from Euclid and Diophantus to problems of foundations in the present state of the art are illustrated by examples.     

A case study of pedagogy of mathematics support tutors without a background in mathematics education

This study investigates the pedagogical skills and knowledge of three tertiary-level mathematics support tutors in a large group classroom setting. This is achieved through the use of video analysis and a theoretical framework comprising Rowland's Knowledge Quartet and general pedagogical knowledge. The study reports on the findings in relation to these tutors’ provision of mathematics support to first and second year undergraduate engineering students and second year undergraduate science students. It was found that tutors are lacking in various pedagogical skills which are needed for high-quality learning amongst service mathematics students (e.g. engineering/science/technology students), a demographic which have low levels of mathematics upon entering university. Tutors teach their support classes in a very fast didactic way with minimal opportunities for students to ask questions or to attempt problems. It was also found that this teaching method is even more so exaggerated in mandatory departmental mathematics tutorials that students take as part of their mathematics studies at tertiary level. The implications of the findings on mathematics tutor training at tertiary level are also discussed.     

Integrating history and philosophy in mathematics education at university level through problem-oriented project work

Through the last three decades several hundred problem-oriented student-directed projects concerning meta-aspects of mathematics and science have been performed in the 2-year interdisciplinary introductory science programme at Roskilde University. Three selected reports from this cohort of project reports are used to investigate and present empirical evidence for learning potentials of integrating history and philosophy in mathematics education. The three projects are: (1) a history project about the use of mathematics in biology that exhibits different epistemic cultures in mathematics and biology. (2) An educational project about the difficulties of learning mathematics that connects to the philosophy of mathematics. (3) A history of mathematics project that connects to the sociology of multiple discoveries. It is analyzed and discussed in what sense students gain first hand experiences with and learn about meta-aspects of mathematics and their mathematical foundation through the problem-oriented student-directed project work.     

Non-Euclidean geometry and Weierstrassian mathematics: The background to Killing's work on Lie algebras

A discussion of the manner in which discoveries in non-Euclidean geometry, combined with the Weierstrassian attitude towards mathematics, led Wilhelm Killing, one of Weierstrass' students, to initiate a research program on foundations of geometry that led to his groundbreaking investigations on the structure of Lie algebras.     

A historic overview of the interplay of theology and philosophy in the arts, mathematics and sciences

The etymology of the word “mathematics” can be traced to Greek roots with meanings such “a thing learned” (mathein is the verb “to have learned”) and, from that, ta mathe^matika, “learnable things” and, “to think or have one’s mind aroused. The natural philosophers of the Renaissance did not draw an explicit distinction between mathematics, the sciences and to an extent the arts. In this paper I explore connections forged by the thinkers of the Renaissance between mathematics, the arts and the sciences, with attention to the nature of the underlying theological and philosophical questions that call for a particular mode of inquiry. Recently Robert Root-Bernstein (2003) introduced the construct of polymathy to suggest that innovative individuals are equally likely to contribute both to the arts and the sciences and either consciously or unconsciously forge links between the two. Several contemporary examples are presented of individuals who pursued multiple fields of research and were able to combine the aesthetic with the scientific. Finally, some possibilities for re-introducing university courses on natural philosophy as a means to integrate mathematics, the arts and the sciences are discussed.     

DNR perspective on mathematics curriculum and instruction, Part I: focus on proving

This is the first in a series of two papers whose goal is to contribute to the debate on a pair of questions: (1) What is the mathematics that we should teach in school? (2) How should we teach it? This paper addresses the first question, and the second paper, to appear in the next issue of ZDM, addresses the second question. The two questions are addressed from a particular theoretical framework, called DNR-based instruction in mathematics. The discussions in the current paper are instantiated mainly in proof-related contexts. The paper offers a definition of mathematics as a union of two categories of knowledge: ways of understanding and ways of thinking. The latter are generalizations of the notions, proof and proof scheme, respectively. The paper also discusses cognitive-epistemological and curricular implications of this definition, focusing mainly on the inevitable production of narrow or faulty mathematical knowledge and the asymmetry in educators’ attention to ways of understanding and ways of thinking.     

Influence of socio-economic background and cultural practices on mathematics education in India: a contemporary overview in historical perspective

The nature, extent and quality of mathematics learning among young children in India cannot be adequately understood without looking at the larger context of education and the social background of the children. Society, including schools, characterized by large inequalities impacts mathematics learning. Beginning with a brief overview of (mathematics) education in India, in historical and sociological perspectives, an appraisal is presented of the need and nature of mathematics learning revealed by field studies in two communities in a deprived rural setting and a low-income urban setting, respectively. While the latter was economically active, the former was much poorer in work and education opportunities, though had richer cultural practices that involved engagement with mathematical riddles, puzzles, folklores and mnemonic tables. The paper discusses the enabling potential of the knowledge resources, including work-context knowledge, which exist in both the communities despite the prevalent deprivations due to disadvantaged conditions. Yet in both situations mathematics learning remains disconnected from formal school mathematics. Factors within SES that possibly have strong bearings on mathematics learning are highlighted which can scaffold stronger integration with curricular and pedagogic practices. Both the groups presented potentially rich contexts for drawing upon everyday mathematical knowledge that can inform effective mathematics learning, which has been inadequately explored in curriculum and instructional design thus far.